Solving (most) of a set of quadratic equalities: Composite optimization for robust phase retrieval

We develop procedures, based on minimization of the composition $f(x) = h(c(x))$ of a convex function $h$ and smooth function $c$, for solving random collections of quadratic equalities, applying our methodology to phase retrieval problems. We show that the prox-linear algorithm we develop can solve phase retrieval problems—even with adversarially faulty measurements—with high probability as soon as the number of measurements $m$ is a constant factor larger than the dimension $n$ of the signal to be recovered. The algorithm requires essentially no tuning—it consists of solving a sequence of convex problems—and it is implementable without any particular assumptions on the measurements taken. We provide substantial experiments investigating our methods, indicating the practical effectiveness of the procedures and showing that they succeed with high probability as soon as $m / n \ge 2$ when the signal is real-valued.

[1]  R. Gerchberg.  A practical algorithm for the determination of phase from image and diffraction plane pictures , 1972 .

[2]  D. Bertsekas.  Approximation procedures based on the method of multipliers , 1977 .

[3]  J R Fienup,et al.  Reconstruction of an object from the modulus of its Fourier transform. , 1978, Optics letters.

[4]  R. Dudley.  Balls in Rk do not cut all subsets of k + 2 points , 1979 .

[5]  B. T. Poljak.  On the Bertsekas' method for minimization of composite functions , 1979 .

[6]  G. Alistair Watson,et al.  First and second order conditions for a class of nondifferentiable optimization problems , 1980, Math. Program..

[7]  J R Fienup,et al.  Phase retrieval algorithms: a comparison. , 1982, Applied optics.

[8]  James V. Burke,et al.  Descent methods for composite nondifferentiable optimization problems , 1985, Math. Program..

[9]  Aharon Ben-Tal,et al.  Lectures on modern convex optimization , 1987 .

[10]  G. Stewart,et al.  Matrix Perturbation Theory , 1990 .

[11]  J. Hiriart-Urruty,et al.  Convex analysis and minimization algorithms , 1993 .

[12]  Michael C. Ferris,et al.  A Gauss—Newton method for convex composite optimization , 1995, Math. Program..

[13]  Jon A. Wellner,et al.  Weak Convergence and Empirical Processes: With Applications to Statistics , 1996 .

[14]  K. Ball.  An Elementary Introduction to Modern Convex Geometry , 1997 .

[15]  L. Trefethen,et al.  Numerical linear algebra , 1997 .

[16]  Peter L. Bartlett,et al.  Rademacher and Gaussian Complexities: Risk Bounds and Structural Results , 2003, J. Mach. Learn. Res..

[17]  M. Ledoux.  The concentration of measure phenomenon , 2001 .

[18]  Trevor Hastie,et al.  The Elements of Statistical Learning , 2001 .

[19]  B. Ripley,et al.  Robust Statistics , 2018, Encyclopedia of Mathematical Geosciences.

[20]  Ken Kreutz-Delgado,et al.  The Complex Gradient Operator and the CR-Calculus ECE275A - Lecture Supplement - Fall 2005 , 2009, 0906.4835.

[21]  R. Balan,et al.  On signal reconstruction without phase , 2006 .

[22]  Andrea Montanari,et al.  Matrix completion from a few entries , 2009, 2009 IEEE International Symposium on Information Theory.

[23]  Emmanuel J. Candès,et al.  Tight Oracle Inequalities for Low-Rank Matrix Recovery From a Minimal Number of Noisy Random Measurements , 2011, IEEE Transactions on Information Theory.

[24]  G. Papanicolaou,et al.  Array imaging using intensity-only measurements , 2010 .

[25]  Joel A. Tropp,et al.  Improved Analysis of the subsampled Randomized Hadamard Transform , 2010, Adv. Data Sci. Adapt. Anal..

[26]  Emmanuel J. Candès,et al.  PhaseLift: Exact and Stable Signal Recovery from Magnitude Measurements via Convex Programming , 2011, ArXiv.

[27]  Roman Vershynin,et al.  Introduction to the non-asymptotic analysis of random matrices , 2010, Compressed Sensing.

[28]  Yonina C. Eldar,et al.  Phase Retrieval: Stability and Recovery Guarantees , 2012, ArXiv.

[29]  Yonina C. Eldar,et al.  Phase Retrieval via Matrix Completion , 2013, SIAM J. Imaging Sci..

[30]  Po-Ling Loh,et al.  Regularized M-estimators with nonconvexity: statistical and algorithmic theory for local optima , 2013, J. Mach. Learn. Res..

[31]  Yang Wang,et al.  Phase retrieval from very few measurements , 2013, ArXiv.

[32]  Dustin G. Mixon,et al.  Saving phase: Injectivity and stability for phase retrieval , 2013, 1302.4618.

[33]  Stephen P. Boyd,et al.  Convex Optimization in Julia , 2014, 2014 First Workshop for High Performance Technical Computing in Dynamic Languages.

[34]  Xiaodong Li,et al.  Solving Quadratic Equations via PhaseLift When There Are About as Many Equations as Unknowns , 2012, Found. Comput. Math..

[35]  Yonina C. Eldar,et al.  Phase Retrieval with Application to Optical Imaging , 2014, ArXiv.

[36]  Martin J. Wainwright,et al.  Statistical guarantees for the EM algorithm: From population to sample-based analysis , 2014, ArXiv.

[37]  Stephen P. Boyd,et al.  Proximal Algorithms , 2013, Found. Trends Optim..

[38]  Shahar Mendelson,et al.  Learning without Concentration , 2014, COLT.

[39]  Stephen P. Boyd,et al.  Block splitting for distributed optimization , 2013, Mathematical Programming Computation.

[40]  Alexandre d'Aspremont,et al.  Phase recovery, MaxCut and complex semidefinite programming , 2012, Math. Program..

[41]  Yuxin Chen,et al.  Solving Random Quadratic Systems of Equations Is Nearly as Easy as Solving Linear Systems , 2015, NIPS.

[42]  Prateek Jain,et al.  Phase Retrieval Using Alternating Minimization , 2013, IEEE Transactions on Signal Processing.

[43]  Xiaodong Li,et al.  Phase Retrieval via Wirtinger Flow: Theory and Algorithms , 2014, IEEE Transactions on Information Theory.

[44]  Yingbin Liang,et al.  Provable Non-convex Phase Retrieval with Outliers: Median TruncatedWirtinger Flow , 2016, ICML.

[45]  Yonina C. Eldar,et al.  Solving Systems of Random Quadratic Equations via Truncated Amplitude Flow , 2016, IEEE Transactions on Information Theory.

[46]  Dmitriy Drusvyatskiy,et al.  Error Bounds, Quadratic Growth, and Linear Convergence of Proximal Methods , 2016, Math. Oper. Res..

[47]  Dmitriy Drusvyatskiy,et al.  Efficiency of minimizing compositions of convex functions and smooth maps , 2016, Math. Program..